Co-existence of states in quantum systems

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Jul 26, 2014 - inherent to many-body system, which might bring about rather .... where the functions are represented by us = u0 +szs +z, uss = szss +2zs, and ...
arXiv:1407.7081v1 [math-ph] 26 Jul 2014

Co-existence of states in quantum systems Yoritaka Iwata School of Science, The University of Tokyo, Hongo 7-3-1, 113-0033 Tokyo E-mail: [email protected] Abstract. Co-existence of different states is a profound concept, which possibly underlies the phase transition and the symmetry breaking. Because of a property inherent to quantum mechanics (cf. uncertainty), the co-existence is expected to appear more naturally in quantummicroscopic systems than in macroscopic systems. In this paper a mathematical theory describing co-existence of states in quantum systems is presented, and the co-existence is classified into 9 types.

1. Introduction The boundary-value problem of nonlinear partial differential equation of elliptic-type: (

−∇2 u − mu − V (u)u = 0 in Ω, u = 0 on ∂Ω,

(1)

is studied, where m is a real number, and Ω ∈ R3 is a closed domain with a sufficiently smooth boundary. The unknown complex function u consists of the unknown state ψ and the reference ¯ state ψ: ¯ u = ψ − ψ, where ψ¯ (corresponding to a generalized concept of the vacuum) is not necessarily a solution of Eq. (1), although the most simplest case ψ¯ = 0 (the simplest vacuum) satisfies Eq. (1). Let a part of the inhomogeneous term V (u), whose spectral set is assumed to be included in a real axis, satisfy ∂u (V (u)u)|u=0 = VL . (2) For the simplicity VL , which corresponds to the signed strength of linearized interaction being independent of u, is assumed to be a real number. As is readily seen, the function u = ψ − ψ¯ = 0 is always a solution of this problem (refer to the trivial solution). In this sense let us imagine a simple case when ψ¯ = 0, and then the emergence of a solution ψ from another solution ψ¯ = 0 is ¯ to true if u 6= 0 is the solution of Eq. (1). Here we seek the non-trivial solution u 6= 0 (ψ 6= ψ) ¯ Eq. (1). The corresponding situation is nothing but the co-existence of different states ψ and ψ. Equation (1) is associated with the stationary problem of nonlinear Schr¨ odinger equations as well as nonlinear Klein-Gordon equations. In the context of Klein-Gordon equations, it √ is possible to associate −m with the mass (if m < 0). Note that the statistical property inherent to many-body system, which might bring about rather interesting physical properties, is not taken into account in order to see the most fundamental properties associated with the co-existence in both nonlinear Schr¨ odinger equations and nonlinear Klein-Gordon equations.

2. Theory describing the co-existence 2.1. Mathematical settings Let X and Y be functional spaces X = W01,2 (Ω) ∩ W 2,2 (Ω), Y = L2 (Ω)

respectively (for mathematical notation, see [1]). An inclusion relation X ⊂ Y is true. For u ∈ X, a mapping f : R1 × X → Y is defined by f (λ, u) := −∇2 u − mu − V (u)u. The original master equation is written by f (λ, u) = 0. Since the trivial solution u = 0 always exists, f (λ, 0) = 0 is satisfied. According to the Sobolev embedding theorem −∇2 is a C 2 mapping from X to Y , where the detail setting of V (u) is necessary to know the regularity of the mapping f . The space W01,2 (Ω) denotes all the functions included in W 1,2 (Ω) satisfying u|∂Ω = 0. 2.2. Linearized analysis Linearized problem is derived. The Fr´echet derivative of f (λ, u) is calculated as fu (λ, 0)[u] = −∇2 u − λu = 0,

(3)

where λ = m + VL . This corresponds to the master equation for the linearized eigen-value problem. It is well known that the linearized problem (with the Dirichlet boundary condition) 2 is solvable. Furthermore it is known that a infinite set of eigen-values {λi }∞ i=0 of −∇ satisfy • 0 < λ0 < λ1 ≤ λ2 ≤ · · ·; • λ0 is a simple eigen-value.

Let the eigen-function corresponding to the eigen-value λ0 be u0 (i.e., −∇2 u0 = λ0 u0 ). First, according to the simple property of the eigen-value λ0 , it is clear that Ker(fu (λ0 , 0)) = {tu0 ; t ∈ R1 }, so that the dimension of Ker(fu (λ0 , 0)) is equal to 1. Second, if there exists a solution v ∈ X for ∇2 v − λ0 v = h with h ∈ Y , then 

R(fu (λ0 , 0)) = h ∈ Y ;

Z



h(x)u(x)dx = 0 , Ω

so that R(fu (λ0 , 0)) is a closed subset of Y with its co-dimension 1 (cf. the Riesz-Schauder theory [1]). Third, it is valid that fuλ (λ0 , 0)[u] = −λ0 u ∈ / R(fu (λ0 , 0)).

(4)

Consequently, according to the bifurcation theory [2, 3], λ = λ0 has been clarified to be a bifurcation point (corresponding to (λ0 , 0) in Fig. 1). Note that only sufficient conditions for the existence of the bifurcation point is presented in the bifurcation theory.

s

s

(iii)

0

0

s

0

s (iv)

0

s (v)

(vi)

0

0

s

s

s (viii)

(vii)

0

s (ii)

(i)

( 

0

0

Figure 1. 9 types of co-existence based on Eqs. (5) and (6): cases (i), (ii), and (iii) appear if µs (0) = 0, cases (iv), (v), and (vi)) appear if µs (0) > 0, cases (vii), (viii), and (ix) appear if µs (0) < 0; cases (i), (iv), and (vii) appear if µss (0) > 0, cases (ii), (v), and (viii) appear if µss = 0, cases (iii), (vi), and (ix) appear if µss < 0. 2.3. Nonlinear analysis Co-existence of different states (i.e., existence of non-trivial solution u 6= 0) is shown. We set a closed interval [−ǫ0 , ǫ0 ] and a C 1 -function λ(s) satisfying λ(0) = λ0 , where s parametrizes the functional space X. Under the three conditions confirmed in Sec. 2.2, let the corresponding solution u be represented by u(λ, s, x) = su0 (λ, x) + sz(λ, s, x), where s is defined on the interval, and z(λ, s, x) is a sufficiently smooth function of s defined on R1 × R1 × X. The function z(λ, s, x) satisfies z(λ, 0, x) = 0 and Z

z(x)u0 (x)dx = 0.



The function u(λ, s, x) satisfies the condition u(λ, 0, x) = 0, which means the existence of the trivial solution. It is useful to define a linear operator A := −∇2 − λ0 , with its domain X, and then it is readily seen that A is a self-adjoint operator in Y . The original equation is written by Au = µ(s)u + (V (u) − VL )u with µ(s) = λ(s) − λ0 , and the linearized problem is written by Au0 = 0. By differentiating the original equation with respect to s, step

by step (Au)s = µs u + µus + ∂s (V (u)u) − VL us

= µs u + µus + (∂s V (u))u + V (u)us − VL us

(Au)ss = µss u + 2µs us + µuss + ∂s2 (V (u)u) − VL uss

= µss u + 2µs us + µuss + (∂s2 V (u))u + 2(∂s V (u))us + V (u)uss − VL uss

(Au)sss = µsss u + 3µss us + 3µs uss + µusss + ∂s3 (V (u)u) − VL usss

= µsss u + 3µss us + 3µs uss + µusss + (∂s3 V (u))u + 3(∂s2 V (u))us + 3(∂s V (u))uss

+V (u)usss − VL usss where the functions are represented by us = u0 + szs + z, uss = szss + 2zs , and usss = szsss + 3zss respectively. The derivatives of the inhomogeneous terms become ∂s V (u) = (∂u V (u)) us ∂s2 V (u) = (∂u2 V (u)) u2s + (∂u V (u)) uss ∂s3 V (u) = (∂u3 V (u)) u3s + 3(∂u2 V (u)) us uss + (∂u V (u)) usss . By taking s = 0, the bi-linear forms become (Au)s |s=0 = µs (0)u|s=0 + µ(0)us |s=0 + ∂u (V (u)u)us |s=0 − VL us |s=0 = VL us |s=0 − VL us |s=0 = 0,

((Au)s |s=0 , u0 ) = 0,

(Au)ss |s=0 = µss (0)u|s=0 + 2µs (0)(u0 + z|s=0 ) + 2µ(0)zs |s=0 + ∂s2 (V (u)u)|s=0 − 2VL zs |s=0 = 2µs (0)u0 + ∂s2 (V (u)u)|s=0 − 2VL zs |s=0 ,

((Au)ss |s=0 , u0 ) = 2µs (0)(u0 , u0 ) + (∂s2 (V (u)u)|s=0 , u0 ) − (2VL zs |s=0 , u0 ), (Au)sss |s=0 = µsss (0)u|s=0 + 3µss (0)(u0 + z|s=0 + 6µs (0)zs |s=0 + 3µ(0)zss |s=0 +∂s3 (V (u)u)|s=0 − 3VL zss |s=0

= 3µss (0)u0 + 6µs (0)zs |s=0 + ∂s3 (V (u)u)|s=0 − 3VL zss |s=0 ,

((Au)sss |s=0 , u0 ) = 3µss (0)(u0 , u0 ) + (6µs (0)zs |s=0 , u0 ) + (∂s3 (V (u)u)|s=0 , u0 ) − 3(VL zss |s=0 , u0 ), where u|s=0 = 0, z|s=0 = 0, and µ(0) = 0 are utilized, as well as Eq. (2). (Auss |s=0 , u0 ) = (uss |s=0 , Au0 ) = 0 due to Au0 = 0. Consequently 2µs (0) = −(∂s2 (V (u)u)|s=0 , u0 ) + 2(VL zs |s=0 , u0 ),

(5)

and the sign of λs (0) = µs (0) is determined by −(∂s2 (V (u)u)|s=0 , u0 ) + 2(VL zs |s=0 , u0 ). In the same manner (Ausss |s=0 , u0 ) = (usss |s=0 , Au0 ) = 0. It leads to 3µss (0) = −(∂s3 (V (u)u)|s=0 , u0 ) − (6µs (0)zs |s=0 , u0 ) + 3(VL zss |s=0 , u0 ),

(6)

and the sign of λss (0) = µss (0) is determined by −(∂s3 (V (u)u)|s=0 , u0 ) − (6µs (0)zs |s=0 , u0 ) + 3(VL zss |s=0 , u0 ). In particular, if λs (0) = µs (0) = 0 is true, the sign of λss (0) is determined by −(∂s3 (V (u)u)|s=0 , u0 )+3(VL zss |s=0 , u0 ). According to Eqs. (5) and (6), the co-existence of states is classified into 9 types (Fig. 1). In Figure 1, around the neighbour of the bifurcation point (λ0 , 0), two solutions co-exist in types (iv) to (ix), while the transition from single-existence to co-existence is described in types (i) and (iii).

Table 1. Systematic analysis for ψ k -interaction theory. Possible classification of co-existence is shown in the column “Type”, where σ = 4η(u0 zs |s=0 , u0 ) − 2η(u20 , u0 )(zs |s=0 , u0 ). k =3 =4 ≥5

∂s V (u) −ηus 0 0

∂s2 V (u) −ηuss −2ηu2s 0

∂s2 (V (u)u)|s=0 −2ηu20 0 0

∂s3 (V (u)u)|s=0 −12ηu0 zs |s=0 −6ηu30 0

µs (0) η(u20 , u0 ) 0 0

µss (0) σ 2η(u30 , u0 ) 0

Type all (i),(ii),(iii) (ii)

3. Application to ψ k -interaction theory If the Lagrangian includes the kth-order nonlinearity in its interaction part (for example, see textbooks of particle physics), the inhomogeneous term of the master equation becomes V (u)u = −ηuk−1 , for integers k ≥ 1, where η is assumed to be a real number. Here VL = 0 and V (u)|s=0 = V (u)|u=0 = 0 are true. The first derivative is ∂u V (u)|u=0 = −(k − 2)ηuk−3 |u=0 for k ≥ 3, so that it is equal to −η for k = 3, and zero for k ≥ 4. The second derivative is ∂u2 V (u)|u=0 = −(k − 2)(k − 3)ηuk−4 |u=0 for k ≥ 4, so that it is equal to zero for k = 3, −2η for k = 4, and zero for k ≥ 5. The third derivative is ∂u3 V (u)|u=0 = −(k − 2)(k − 3)(k − 4)ηuk−5 |u=0 for k ≥ 5, so that it is equal to zero for k ≤ 4, −6η for k = 5, and zero for k ≥ 6. Results are summarized in Table 1. In case of k = 4 (ψ 4 -interaction theory), the non-trivial solution corresponds to type (i) of Fig. 1 if η > 0, to type (ii) if η = 0, and to type (iii) if η < 0. In particular when η > 0, the co-existence emerges only if m > λ0 (cf. spontaneous symmetry breaking). If there is no interaction (free particle condition: η = 0), µs (0) = µss (0) = 0 is true, and the co-existence is classified into type (ii). If the interaction is linear (V (u) = VL 6= 0; ψ 2 -interaction theory), the derivatives are ∂u V (u)|u=0 = ∂u2 V (u)|u=0 = ∂u3 V (u)|u=0 = 0, so that ∂s V (u) = ∂s2 V (u) = ∂s3 V (u) = 0. It leads to ∂s2 (V (u)u)|s=0 = VL uss |s=0 and ∂s3 (V (u)u)|s=0 = VL usss |s=0 so that µs (0) = −(VL zs |s=0 , u0 ) + (VL zs |s=0 , u0 ) = 0 and µss (0) = −(VL zss |s=0 , u0 ) + (VL zss |s=0 , u0 ) = 0 follows. The co-existence is classified into type (ii). As a result the nonlinearity can be identified by the classification other than type (ii). Acknowledgments This work was supported by HPCI Strategic Programs for Innovative Research Field 5 “The origin of matter and the universe”. The author is grateful to Prof. Emeritus Dr. Hiroki Tanabe for reading the manuscript. References [1] K. Yosida, Functional Analysis, Sixth Edition, Springer-Verlag, Berlin Heidelberg New York, 1980. [2] M. G. Crandall and P. H. Rabinowitz, J. Funct. Anal. (1971) 8 321. [3] M. G. Crandall and P. H. Rabinowitz, Arc. Rational Mech. Anal. 21, XI (1973) 52 2, 161.